Free Energy of Granular Materials in Static Equilibrium

1979 ◽  
Vol 46 (4) ◽  
pp. 944-945 ◽  
Author(s):  
M. Shahinpoor ◽  
G. Ahmadi
Keyword(s):  
Free Energy ◽  
Granular Materials ◽  
Continuum Theory ◽  
Gravitational Effect ◽  
Experimental Results ◽  
Static Equilibrium ◽  
The Continuum

We employ the continuum theory of granular materials due to Goodman and Cowin and some experimental results due to P. G. Nutting to arrive at a functional from for the free energy of granular materials in static equilibrium. The results obtained indicate the dominance of gravitational effect, modify and enlarge the results previously obtained by J. T. Jenkins.

A unified formulation for plasticity models of granular and other materials

1995 ◽  
Vol 450 (1938) ◽  
pp. 37-49 ◽  
Keyword(s):  
Granular Materials ◽  
Potential Model ◽  
Continuum Theory ◽  
Potential Models ◽  
Special Cases ◽  
Plastic Potential ◽  
Physically Based ◽  
Relationship Of ◽  
The Relationship ◽  
The Continuum

A general physically based rule for the kinematics of planar deformation of granular materials is presented and a pair of kinematic equations governing the velocity field is obtained. The method results in a unified derivation of the equations for the double sliding, free rotating model, the double shearing model and the plastic potential model for granular materials. The formulation of the double sliding, free rotating model for the kinematics of granular materials presented here is consistent with the continuum theory of constitutive equations and demonstrates the relationship of this model both to the double shearing model and to the plastic potential models. Expressions for the rate of working of the stresses are presented. A constitutive equation, expressed in terms of planar tensor quantities, is also presented that has the three models as special cases.

Thermodynamics of Flowing Systems: with Internal Microstructure

1994 ◽  
Author(s):  
Antony N. Beris ◽  
Brian J. Edwards
Keyword(s):  
Free Energy ◽  
Theoretical Study ◽  
Flow Behavior ◽  
Irreversible Thermodynamics ◽  
Hamiltonian Mechanics ◽  
Complex Interplay ◽  
Microscopic Models ◽  
Flow Phenomena ◽  
The Subject ◽  
The Continuum

This much-needed monograph presents a systematic, step-by-step approach to the continuum modeling of flow phenomena exhibited within materials endowed with a complex internal microstructure, such as polymers and liquid crystals. By combining the principles of Hamiltonian mechanics with those of irreversible thermodynamics, Antony N. Beris and Brian J. Edwards, renowned authorities on the subject, expertly describe the complex interplay between conservative and dissipative processes. Throughout the book, the authors emphasize the evaluation of the free energy--largely based on ideas from statistical mechanics--and how to fit the values of the phenomenological parameters against those of microscopic models. With Thermodynamics of Flowing Systems in hand, mathematicians, engineers, and physicists involved with the theoretical study of flow behavior in structurally complex media now have a superb, self-contained theoretical framework on which to base their modeling efforts.

scholarly journals Debye-Hückel Free Energy of an Electric Double Layer with Discrete Charges Located at a Dielectric Interface

Membranes ◽  
2021 ◽  
Vol 11 (2) ◽  
pp. 129
Author(s):  
Guilherme Volpe Bossa ◽  
Sylvio May
Keyword(s):  
Free Energy ◽  
Double Layer ◽  
Electric Double Layer ◽  
Low Dielectric Constant ◽  
Surface Charges ◽  
Dielectric Interface ◽  
Charge Densities ◽  
Low Dielectric ◽  
Poisson Boltzmann ◽  
The Continuum

Poisson–Boltzmann theory provides an established framework to calculate properties and free energies of an electric double layer, especially for simple geometries and interfaces that carry continuous charge densities. At sufficiently small length scales, however, the discreteness of the surface charges cannot be neglected. We consider a planar dielectric interface that separates a salt-containing aqueous phase from a medium of low dielectric constant and carries discrete surface charges of fixed density. Within the linear Debye-Hückel limit of Poisson–Boltzmann theory, we calculate the surface potential inside a Wigner–Seitz cell that is produced by all surface charges outside the cell using a Fourier-Bessel series and a Hankel transformation. From the surface potential, we obtain the Debye-Hückel free energy of the electric double layer, which we compare with the corresponding expression in the continuum limit. Differences arise for sufficiently small charge densities, where we show that the dominating interaction is dipolar, arising from the dipoles formed by the surface charges and associated counterions. This interaction propagates through the medium of a low dielectric constant and alters the continuum power of two dependence of the free energy on the surface charge density to a power of 2.5 law.

Confirming the continuum theory of dynamic brittle fracture for fast cracks

Nature ◽  
10.1038/16891 ◽  
1999 ◽  
Vol 397 (6717) ◽  
pp. 333-335 ◽  
Author(s):  
Eran Sharon ◽  
Jay Fineberg
Keyword(s):  
Brittle Fracture ◽  
Continuum Theory ◽  
Dynamic Brittle Fracture ◽  
The Continuum

scholarly journals Matrix integrals & finite holography

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Dionysios Anninos ◽  
Beatrix Mühlmann
Keyword(s):  
Critical Exponents ◽  
Continuum Theory ◽  
Point Of View ◽  
Leading Order ◽  
Minimal Models ◽  
Point Evaluation ◽  
Brst Cohomology ◽  
Matrix Integrals ◽  
The Matrix ◽  
The Continuum

Abstract We explore the conjectured duality between a class of large N matrix integrals, known as multicritical matrix integrals (MMI), and the series (2m − 1, 2) of non-unitary minimal models on a fluctuating background. We match the critical exponents of the leading order planar expansion of MMI, to those of the continuum theory on an S2 topology. From the MMI perspective this is done both through a multi-vertex diagrammatic expansion, thereby revealing novel combinatorial expressions, as well as through a systematic saddle point evaluation of the matrix integral as a function of its parameters. From the continuum point of view the corresponding critical exponents are obtained upon computing the partition function in the presence of a given conformal primary. Further to this, we elaborate on a Hilbert space of the continuum theory, and the putative finiteness thereof, on both an S2 and a T2 topology using BRST cohomology considerations. Matrix integrals support this finiteness.

On the Motion of Granular Materials Due to Shear

2000 ◽  
Author(s):  
Mehrdad Massoudi ◽  
Tran X. Phuoc
Keyword(s):  
Finite Difference ◽  
Granular Materials ◽  
Continuum Model ◽  
Theory Model ◽  
Governing Equations ◽  
Flat Plates ◽  
Material Parameters ◽  
Finite Difference Technique ◽  
Linear Differential ◽  
The Continuum

Abstract In this paper we study the flow of granular materials between two horisontal flat plates where the top plate is moving with a constant speed. The constitutive relation used for the stress is based on the continuum model proposed by Rajagopal and Massoudi (1990), where the material parameters are derived using the kinetic theory model proposed by Boyle and Massoudi (1990). The governing equations are non-dimensionalized and the resulting system of non-linear differential equations is solved numerically using finite difference technique.

Experimental evidence for the need of thermodynamic considerations in modelling of anaerobic environmental bioprocesses

1997 ◽  
Vol 36 (10) ◽  
pp. 109-115 ◽  
Author(s):  
Choon-Yee Hoh ◽  
Ralf Cord-Ruwisch
Keyword(s):  
Free Energy ◽  
Experimental Evidence ◽  
Positive Reaction ◽  
Rate Equation ◽  
Dynamic Equilibrium ◽  
Reaction Rates ◽  
Product Inhibition ◽  
Experimental Results ◽  
Biological Processes ◽  
Laws Of Thermodynamics

For modeling of biological processes that operate close to the dynamic equilibrium (eg. anaerobic processes), it is critical to prevent the prediction of positive reaction rates when the reaction has already reached dynamic equilibrium. Traditional Michaelis-Menten based models were found to violate the laws of thermodynamics as they predicted positive reaction rates for reactions that were endergonic due to high endproduct concentrations. The inclusion of empirical “product inhibition factors” as suggested by previous work could not prevent this problem. This paper compares the predictions of the Michaelis-Menten Model (with and without product inhibition factors) and the Equilibrium Based Model (which has a thermodynamic term introduced into its rate equation) with experimental results of reactions in anaerobic bacterial environments. In contrast to the Michaelis-Menten based models that used traditional inhibition factors, the Equilibrium Based Model correctly predicted the nature and the degree of inhibition due to endproduct accumulation. Moreover, this model also correctly predicted when reaction rates must be zero due to the free energy change of the conversion reaction being zero. With these added advantages, the Equilibrium Based Model thus seemed to provide a scientifically correct and more realistic basis for a variety of models that describe anaerobic biosystems.

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